Application of Derivatives Test

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Application of Derivatives Test - 44

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Question 1
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If the tangent at $$(x_{1}, y_{1})$$ to the curve $$x^{3}+y^{3}=a^{3}$$ meets the curve again at $$(x_{2}, y_{2})$$ then

A
$$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=-1$$

B
$$\dfrac{x_{2}}{y_{1}}+\dfrac{x_{1}}{y_{2}}=-1$$

C
$$\dfrac{x_{1}}{x_{2}}+\dfrac{y_{1}}{y_{2}}=-1$$

D
$$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=1$$

Solution

Given $$x^3+y^3=a^3$$
The derivative is
$$\dfrac{dy}{dx}=-\dfrac{x^2}{y^2}.....(1)$$
Therefore, slope of tangent at $$(x_1,y_1)$$ is
$$-\dfrac{x_1^2}{y_1^2}.........(2)$$
The tangent passes through $$(x_2,y_2)$$, therefore the slope of tangent is also given by
$$\dfrac{y_2-y_1}{x_2-x_1}.......(3)$$
Comparing the two slope equations we get
$$\dfrac{y_2-y_1}{x_2-x_1}=-\dfrac{x_1^2}{y_1^2}$$

$$\dfrac{y_2^3-y_1^3}{x_2^3-x_1^3}\times\dfrac{x_1^2+x_1x_2+x_2^2}{y_1^2+y_1y_2+y_2^2}=-\dfrac{x_1^2}{y_1^2}$$

$$-\dfrac{x_1^2+x_1x_2+x_2^2}{y_1^2+y_1y_2+y_2^2}=-\dfrac{x_1^2}{y_1^2}$$

$$x_1^2y_1^2+x_1x_2y_1^2+x_2^2y_1^2=x_1^2y_1^2+x_1^2y_1y_2+x_1^2y_2^2$$

$$x_1x_2y_1^2+x_2^2y_1^2=x_1^2y_1y_2+x_1^2y_2^2$$

$$x_1^2y_2^2-x_2^2y_1^2=x_1x_2y_1^2-x_1^2y_1y_2$$

$$(x_1y_2-x_2y_1)(x_1y_2+x_2y_1)=x_1y_1(x_2y_1-x_1y_2)$$

$$x_1y_2+x_2y_1=-x_1y_1$$

$$\dfrac{x_2}{x_1}+\dfrac{y_2}{y_1}=-1$$
Question 2
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The ordinate of all points on the curve $$y=\dfrac{1}{2\sin^{2}x+3\cos^{2}x}$$ where the tangent is horizontal, is

A
Always equal to $$\dfrac{1}{2}$$

B
Always equal to $$\dfrac{1}{3}$$

C
$$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even or an odd integer.

D
$$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even integer

Solution

$$y=\dfrac{1}{2sin^{2}x+3cos^{2}x}=\dfrac{1}{2+cos^{2}x}$$
$$tan\theta =\dfrac{dy}{dx}= \dfrac{-2sinx\, cos x}{(2+cos^{2}x)^{2}}=0$$
$$\therefore sin\, x=0$$ or $$cos\, x= 0$$
If $$sin\, x=0, y= \dfrac{1}{3}$$, if $$cos\, x=0, y= \dfrac{1}{2}$$
Question 3
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The curve given by $$x + y = {e^{xy}}$$ has an tangents parallel to the y-axis at the point

A
$$(0,1)$$

B
$$(1,0)$$

C
$$(1,1)$$

D
$$(0,0)$$

Solution

$$\dfrac{dy}{dx}$$(differentiate the terms)
$$1+\dfrac{dy}{dx}=e^{xy}\left(x\dfrac{dy}{dx}+y\right)$$
$$\Rightarrow \dfrac{dy}{dx}=\dfrac{ye^{xy}-1}{1-xe^{xy}}$$
Check by option
$$\dfrac{dy}{dx}\left|_{(0, 1)}\right.=\dfrac{1-1}{1-0}=\dfrac{0}{1}=0$$ Not coming infinity
$$\displaystyle\dfrac{dy}{dx}\left|_{(1, 0)}\right.=\dfrac{-1}{0}=\infty$$
This point satisfy
$$\therefore$$ option B$$(1, 0)$$.
Question 4
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The slope of normal to the curve y= log (logx) at x = e is

A
e

B
-e

C
$$\frac{1}{e}$$

D
-$$\frac{1}{e}$$

Solution

$$\begin{array}{l} \frac { d }{ { dx } } \log \left( { \log x } \right) \\ Substitute\, u=\log x \\ =\dfrac { d }{ { du } } \log u\frac { d }{ { dx } } \log x \\ =\dfrac { 1 }{ u } \left( { \frac { 1 }{ x } } \right) \\ =\dfrac { 1 }{ { \log x } } \left( { \frac { 1 }{ x } } \right) \\ Substitute\, x=e \\ =\dfrac { 1 }{ { \log e } } \left( { \dfrac { 1 }{ e } } \right) \\ =\dfrac { 1 }{ e } \end{array}$$
Question 5
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Number of possible tangents to the curve $$y = \cos \left( {x + y} \right), - 3\pi \leqslant x \leqslant 3\pi $$, that are parallel to the line $$x + 2y = 0$$, is

A
1

B
2

C
3

D
4

Solution
Question 6
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The normal to the curve, $${x}^{2}+2xy-{3y}^{2}=0,\ at\left (1,1\right)$$:

A
Meets the curve, again in the fourth quadrant

B
Does not meet the curve again

C
Meets the curve again in the second quadrant

D
Meets the curve again in the third quadrant

Solution
Question 7
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Number of tangents drawn from the point $$\left (-1/2,0\right)$$ to the curve $$y={e}^{x}$$. (Here { } denotes fractional part function ).

A
$$2$$

B
$$1$$

C
$$3$$

D
$$4$$

Solution

Let $$A$$ be the point of contact of a tangent of $${e}^{x}$$ passing through $$(-\cfrac{1}{2},0)$$
$$\Rightarrow$$ $${ \left| \cfrac { dy }{ dx } \right| }_{ \lambda }={ e }^{ \lambda }$$
Equation of tangent $$\quad y-{ e }^{ \lambda }={ e }^{ \lambda }(x-\lambda )\quad $$
$$x=-\cfrac { 1 }{ 2 } ,y=0$$
$$-{ e }^{ \lambda }={ e }^{ \lambda }\left( -\cfrac { 1 }{ 2 } -\lambda \right) \Rightarrow { e }^{ \lambda }\lambda =\cfrac { { e }^{ \lambda } }{ 2 } $$
either
$$\quad { e }^{ \lambda }=0$$ or $$\lambda =+\cfrac { 1 }{ 2 } $$
$$\quad y{ e }^{ \lambda }=0\Rightarrow y=0\quad \rightarrow$$ xaxis
$$\quad \lambda =\cfrac { 1 }{ 2 } \Rightarrow y-{ e }^{ \cfrac { 1 }{ 2 } }={ e }^{ \cfrac { 1 }{ 2 } }\left( x-\cfrac { 1 }{ 2 } \right) $$
2 solutions or 2 distinct tangents
Question 8
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If the tangent at P of the curve $$y^2=x^3$$ intersects the curve again at Q and the straight lines OP, OQ ma angles $$\alpha, \beta$$ with the x-axis where 'O' is the origin then $$\tan\alpha/\tan\beta$$ has the value equal to?

A
$$-1$$

B
$$-2$$

C
$$2$$

D
$$\sqrt{2}$$

Solution

Let $$P$$ be $$(x_{1}, y_{1})$$ and $$Q$$ be $$(x_{2}, y_{2})$$
equation of tangent $$(x_{2}, y_{2})$$, so $$\dfrac{y_{0}-y_{1}}{x_{2}-x_{1}}= \dfrac{3x_{1}^{2}}{2y_{1}}$$
need to find out $$\dfrac{\tan \alpha}{ \tan \beta} = \dfrac{\dfrac{y_{1}}{x_{1}}}{y_{2}/x_{2}} = \dfrac{y_{1}x_{2}}{x_{1} y_{2}}$$
$$\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}= \dfrac{3x_{1}^{2}}{2y_{1}}$$
$$2y_{1}y_{2}-2y^{2}= 3x^{2}_{1}x_{2}-3x_{1}^{3}$$
$$2y_{1}y_{2}-2y_{1}^{2}= 3x_{1}^{2}x_{2}-3y^{2}$$
$$2y_{1}y_{2}+y_{1}^{2}= 3x_{1}^{2} x_{2}$$
$$y_{1}^{3} (2y_{2}+y_{1})^{3}= 27 x_{1}^{6} x^{3}_{2}$$
$$y_{1}^{3} (2y_{2}+ y_{1})^{3}= 27 y_{1}^{4} y_{2}^{3}$$
$$(2y_{2}+y_{1})^{3} - 27 y_{1} y_{2}$$
$$8 y_{2}^{3} - 15 y_{2}^{2} y_{1}+ 6y_{2} y^{2}_{1}+ y_{1}^{3}=0$$
$$(y_{2}-y_{1})^{2} (8y_{2}+ y_{1})=0$$
$$8y_{2}=-y_{1} \Rightarrow 64 y_{2}^{2} = y_{1}^{2}$$
$$64 x_{2}^{3} = x_{1}^{3} \Rightarrow 4 x_{2} = x_{1}$$
$$\dfrac{\tan \alpha}{\tan \beta}= \dfrac{y_{1}x_{2}}{x_{1}y_{2}}$$
$$= \dfrac{-8y_{2} x_{2}}{4x_{2} y_{2}} = -2$$
Question 9
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A curve C has the property that if the tangent drawn at any point 'P' on C meets the coordinate axes at A and B, and P is midpoint of AB. If the curve passes through the point $$(1, 1)$$ then the equation of the curve is?

A
$$xy=2$$

B
$$xy=3$$

C
$$xy=1$$

D
$$xy=4$$

Solution

By option verification passing through $$(1,1)$$
$$xy=1$$
Question 10
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Let $$f$$ be continuous and differentiable function such that $$f(x)$$ and $$f^{'}(x)$$ have opposite sign everywhere. Then

A
$$f$$ is increasing

B
$$f$$ is decreasing

C
$$|f|$$ is non-monotonic

D
none of the above

Solution